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Questions and Answers
What is the multiplicative identity?
What is the multiplicative identity?
1
What is the additive identity?
What is the additive identity?
0
Every real number has a decimal representation.
Every real number has a decimal representation.
True
What does the symbol lal represent?
What does the symbol lal represent?
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Which of these options are correct? (Select all that apply)
Which of these options are correct? (Select all that apply)
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What is the formula for the distance between two points on a real number line?
What is the formula for the distance between two points on a real number line?
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Which of these statements are true about 'a' and 'b' in Python? (Choose all that apply)
Which of these statements are true about 'a' and 'b' in Python? (Choose all that apply)
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Match the following features with the correct operation:
Match the following features with the correct operation:
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What is a proposition in logic?
What is a proposition in logic?
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Study Notes
Real Numbers
- Real numbers include natural numbers (1, 2, 3...), integers (-3, -2, -1, 0, 1, 2...), rational numbers (fractions) and irrational numbers (e.g., √2, π).
- Every real number has a decimal representation.
Properties of Real Numbers
- Commutative: a + b = b + a and ab = ba
- Associative: (a + b) + c = a + (b + c) and (ab)c = a(bc)
- Distributive: a(b + c) = ab + ac
- Additive identity: a + 0 = a
- Multiplicative identity: a × 1 = a
- Absolute value: |a| = a if a ≥ 0, |a| = -a if a < 0. The absolute value represents the distance from zero on the number line.
Fractions
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Operations on fractions
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Properties of fractions
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Commutative: a/b = b/a -Associative: (a/b)/c=a/(bc)
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Distributive: a(b/c) = (a/1)(b/c)=ab/c
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Additive indentity:(a/b)+0=(a/b)
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Multiplicative indentity: (a/b)(1)=(a/b)
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Properties of Absolute value
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|a| ≥ 0
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|a| = |-a|
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|ab| = |a||b|
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|a/b| = |a|/|b|
Sets
- A set is a collection of distinct objects.
- Natural numbers: {1, 2, 3...}
- Integers: {..., -3, -2, -1, 0, 1, 2, 3...}
- Rational numbers: numbers that can be expressed as a fraction p/q where p and q are integers.
- Irrational numbers: numbers that cannot be expressed as a fraction p/q.
- Real numbers: the set of all rational and irrational numbers.
- Operations on sets: union, intersection, difference, complement
- Operations on multisets (sets containing repeated elements)
Sequences
- A sequence is an ordered list of numbers.
- Arithmetic progression: aₙ = a₁ + (n-1)d
- Geometric progression: aₙ = a₁r^(ₙ⁻¹), r = common ratio
- Recurrence relations: a rule defining a sequence in terms of earlier terms. (e.g., Fibonacci sequence).
Propositional Logic
- Propositional logic deals with statements that can be true or false.
- Connectives: negation, conjunction, disjunction, implication, biconditional.
Functions
- A function maps elements from a set (domain) to a set (co-domain).
- Types of functions: injective, surjective, bijective.
- Function composition: applying one function to the result of another.
- Special functions: floor, ceiling, factorial functions.
Graphs
- Graphs are structures that consist of vertices and edges connecting pairs of vertices.
- Types of graphs (simple graphs, multigraphs, directed graphs, etc.).
- Graph terminology: vertex, edge, degree, path, cycle.
- Applications of graphs: modeling networks, social relations.
Probability
- Probabilities measure the likelihood of events.
- Discrete probability deals with probabilities of discrete random variables.
- Probability rules: complementary events, union of events, conditional probability, independence.
Matrices
- Matrices are rectangular arrays of numbers.
- Matrix operations: addition, subtraction, scalar multiplication, matrix multiplication.
- Determinant of a matrix: a scalar value that describes the matrix's properties.
- Applications of matrices: representing linear equations, solving systems of equations.
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